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# The Möbius function

For any positive integer $ n $, the Möbius function $ \mu(n) $ is
defined as follows:

$$ \mu(1) = 1; $$

If $ n > 1, $ write $ n = p_1^{a_1} \dots p_2^{a_k} $ (prime
factorization). Then

\begin{align*}
  \mu(n) & = (-1)^k \text{ if } a_1 = a_2 = \dots = a_k = 1, \\
  \mu(n) & = 0 \text{ otherwise}.
\end{align*}

If $ n \ge 1, $ we have

$$
  \sum_{d \mid n} \mu(d) =
  \begin{cases}
    1 & \text{ if } n = 1, \\
    0 & \text{ if } n > 1.
  \end{cases}
$$
